Fractional Fuzzy Adaptive Methodology for Fractional-order Non-Affine Nonlinear Systems: Application to Gyroscope

Document Type : Research Article

Authors

1 Department of Electrical Engineering, University of Qom, Qom, Iran.

2 Department of Mathematics, University of Qom, Qom, Iran.

3 Department of Electrical Engineering, University of Qom, Qom, IRAN

Abstract

This study employs a fractional fuzzy adaptive methodology to design procedures for fractional-order non-affine nonlinear systems. The significant evolution of fractional-order calculus in science and engineering has made this area one of the most widespread fields, particularly in control engineering. Fractional-order fuzzy adaptive controller (FAC) has involved numerous scientists to improve appropriate controllers for non-affine nonlinear systems because of: 1) reconfigurable framework, the performance of the FAC is superior to that of the fuzzy controllers, 2) using the experts’ data, FAC can apply the expert knowledge in the controller procedure rather than adaptive ones, and 3) enhancement of the controller routine instead of the integer-order one. In addition, this approach can control nominal systems in the presence of both external disturbances and uncertainties. The fractional-order adaptation laws are developed to guarantee the stability of the closed-loop system using a fractional-order Lyapunov approach. Unlike other research that focuses on fractional-order affine nonlinear systems, our approach specifically addresses fractional-order nonaffine nonlinear systems. Finally, the performance of the proposed methodology on chaotic systems, a gyroscope, and an inverted pendulum indicates the capability of the proposed scheme.

Keywords

Main Subjects

References

[1] LX. Wang, "A Course in Fuzzy Systems and Control", Prentice Hall PTR, 1997.
[2] LX. Wang, JM. Mendel, "Fuzzy basis function, universal approximation, and orthogonal least square learning", IEEE Trans Neural Network 3(5), 807–814, 1993.
[3] HJ. Zimmermann, "Fuzzy Set Theory and Its Application, Kluwer Academic Publishers, 1996.
[4] B. Vinagre and V. Feliu, " Modeling and control of dynamic systems using fractional calculus: Application to electrochemical processes and flexible structures", in Proc. 41st IEEE Decision and Control Conference, 214-239, 2002.
[5] R. L. Magin, "Fractional calculus in bioengineering", Begell House, Redding, publisher, 2006.
[6] J. Sabatier, M. Aoun, A. Oustaloup, G. Grégoire, F. Ragot, and P. Roy, "Fractional system identification for lead acid battery state of charge estimation", Signal processing, vol. 86, pp. 2645- 2657, 2006.
[7] R. Hilfer, "Applications of fractional calculus in physics", World Scientific publisher, 2000.
[8] C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu-Batlle, "Fractional-order systems and controls: fundamentals and applications", Springer Science & Business Media publisher, 2010.
[9] A. Oustaloup, X. Moreau, and M. Nouillant, "The CRONE suspension," Control Engineering Practice 4(1), 1101-1108, 1996.
[10] M. Zamani, M. Karimi-Ghartemani, N. Sadati, and M. Parniani, "Design of a fractional order PID controller for an AVR using particle swarm optimization," Control Engineering Practice 17(1), 1380-1387, 2009.
[11] N. Aguila-Camacho and M. A. Duarte-Mermoud, "Fractional adaptive control for an automatic voltage regulator," ISA transactions 52(1), 807-815, 2013.
[12] Y. He and R. Gong, "Application of fractional-order model reference adaptive control on industry boiler burning system," in 2010 International Conference on Intelligent Computation, Changsha, China, 2010.
[13] L. Jun-Guo, "Chaotic dynamics and synchronization of fractional-order Genesio–Tesi systems," Chinese Physics 14(8), 1517, 2005.
[14] C. Li and G. Chen, "Chaos and hyperchaos in the fractional-order Rössler equations," Physica A: Statistical Mechanics and its Applications 31(3), 55-61, 2004.
[15] J. G. Lu, "Chaotic dynamics of the fractional-order Lü system and its synchronization," Physics Letters A 354(1), 305-311, 2006.
[16] Q. Yang and C. Zeng, "Chaos in fractional conjugate Lorenz system and its scaling attractors," Communications in Nonlinear Science and Numerical Simulation 15(2), 4041-4051, 2010.
[17] MR. Rahimi Khoygani, R. Ghasemi, "Neural estimation using a stable discrete-time MLP observer for a class of discrete-time uncertain nonlinear systems", Nonlinear Dynamics 84(4), 2517-2533, 2016.
[18] MR. Rahimi Khoygani, R. Ghasemi, A.R. Vali, "Intelligent nonlinear observer design for a class of nonlinear discrete-time flexible joint robot", Intelligent Service Robotics 8(1), 45–56, 2015.
[19] A. Sharafian, R. Ghasemi, "A novel terminal sliding mode observer with RBF neural network for a class of nonlinear systems", International Journal of Systems, Control and Communications 9(4), 369-385, 2019.
[20] F. Shahbazi, R. Ghasemi, M. Mahmoodi, "Fractional Nonsingular Terminal Sliding Mode Controller Design for the Special Class of Nonlinear Fractional-order Chaotic Systems", International Journal of Smart Electrical Engineering 11(2), 83-88, 2021.
[21] M. Ö. Efe, "Fractional fuzzy adaptive sliding mode control of a 2-DOF direct-drive robot arm", Systems, IEEE Transactions on Man and Cybernetics Part B: Cybernetics 38(1), 1561-1570, 2008.
[22] T.-C. Lin, T.-Y. Lee, and V. E. Balas, "Adaptive fuzzy sliding mode control for synchronization of uncertain fractional order chaotic systems," Chaos, Solitons & Fractals 44(2), 791-801, 2011.
[23] T.-C. Lin, C.-H. Kuo, T.-Y. Lee, and V. E. Balas, "Adaptive fuzzy H∞tracking design of SISO uncertain nonlinear ractional order time-delay systems," Nonlinear Dynamics 69(3), 1639- 1650, 2012.
[24] T.-C. Lin and T.Y. Lee, "Chaos synchronization of uncertain fractionalorder chaotic systems with time delay based on adaptive fuzzy sliding mode control," IEEE Transactions on Fuzzy Systems 19(2), 623-635, 2011.
[25] M. P. Aghababa, Comments on “Adaptive fuzzy H∞ tracking design of SISO uncertain nonlinear fractional order time-delay systems”, [Nonlinear Dyn. 69 (2012) 1639–1650]. Nonlinear Dynamics, 70(4), 2511-2513, 2012.
[26] S. Tong, H.-X. Li, and W. Wei, "Comments on Direct adaptive fuzzyneural control with state observer and supervisory controller for unknown nonlinear dynamical systems", IEEE Transactions on Fuzzy Systems 11(2), 703-705, 2003.
[27] T.-C. Lin, C.-H. Kuo, and V. E. Balas, "Uncertain fractional order chaotic systems tracking design via adaptive hybrid fuzzy sliding mode control," International Journal of Computers Communications & Control 6(2), 418-427, 2011.
[28] Y. Li, YQ. Chen, I. Podlubny, "Mittag–Leffler stability of fractional-order non-linear dynamic systems", Automatica 45(8), 1965-1969, 2009.
[29] N. Aguila-Camacho, M. Duarte-Mermoud, J. Gallegos, "Lyapunov Functions for Fractional Order Systems", Communications in Nonlinear Science and Numerical Simulation 19(9), 2951-2957, 2014.
[30] P. Jafari, M. Teshnehlab, and M. Tavakoli-Kakhki, "Adaptive Type-2 Fuzzy System for Synchronization and Stabilization of Chaotic Nonlinear Fractional Order Systems," IET Control Theory & Applications 12(3), 183-193, 2017.
[31] P. Jafari, M. Teshnehlab, and M. Tavakoli-Kakhki, "Synchronization and stabilization of fractional order nonlinear systems with adaptive fuzzy controller and compensation signal," Nonlinear Dynamics 90, 1037-1052, 2017.
[32] L.X. Wang, "Adaptive Fuzzy Systems and Control: Design and Stability Analysis", Prentice-Hall, Inc., 1994.
[33] A. Kumar, V. Kumar, “An interval type-2 fractional order fuzzy logic controller employed to an uncertain nonlinear inverted pendulum”, 14th IEEE India Council International Conference, 2017.
[34] L. Chen, J. Fang, “Adaptive continuous sliding mode control for fractional order systems with uncertainties and unknown control gain”, Automation and Systems 20, 1509- 1520, 2022.
[35] N. Ullah, S. Han, MI. Khattak, “Adaptive fuzzy fractional order sliding mode controller for a class of dynamical systems with uncertainty”, Transactions of the Institute of Measurement and Control 38, 402- 413, 2016.
[36] S. Mirzajani, MP. Aghababa, A. Heydari, “Adaptive T- S fuzzy control design for fractional order systems with parametric uncertainty and input constraint”, Fuzzy Sets and Systems 365, 22- 39, 2019.
[37] X. Qin, S. Liu, “Adaptive fuzzy synchronization of uncertain fractional order chaotic systems with different structures and time delays”, Advances in Difference Equations 174, 2019.
[38] X. Hu, Q. Song, M. Ge, R. Li, “Fractional order adaptive fault tolerant control for a class of general nonlinear systems”, Nonlinear Dynamics 101, 379- 392, 2020.
[39] YX. Li, QY. Wang, S. Tong, “Fuzzy adaptive fault tolerant control of fractional order nonlinear systems”, IEEE Transactions on Systems, Man, and Cybernetics: Systems 51, 1372- 1379, 2021.
[40] A. Mohammadzadeh, O. Kaynak, “A novel fractional order fuzzy control method based on immersion and invariance approach”, Applied Soft Computing 88, 2020.
[41] O. Mofid, S. Mobayen, “Adaptive synchronization of fractional order quadratic chaotic flows with nonhyperbolic equilibrium”, Journal of vibration and control 24, 2018.
[42] A. Sharafian, A. Ali, I. Ullah, T.R. Khalifa, X. Bai, L. Qiu, “Fuzzy adaptive control for consensus tracking in multiagent systems with incommensurate fractional-order dynamics: Application to power systems”, Information Sciences 689(1): 121455, 2025.
[43] M.A.Z. Tajrishi, A. Akbarzadeh Kalat, “Fast finite time fractional-order robust-adaptive sliding mode control of nonlinear systems with unknown dynamics”, Journal of Computational and Applied Mathematics 438: 115554, 2024.
Journal of Modeling and Simulation in Electrical and Electronics Engineering
Volume 5, Issue 4 - Serial Number 22
December 2025
Pages 25-34

Files

History

  • Receive Date: 13 July 2025
  • Revise Date: 06 December 2025
  • Accept Date: 27 December 2025

Share

How to cite

Statistics